MIMO Toolbox

8/11/2019 MIMO Toolbox

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MIMO Toolbox

For Use with MATLAB

Oskar Vivero


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About the toolbox

The MIMO Toolbox is a collection of MATLABfunctions and a GUI. Its purpose is to

complement the Control Toolbox for MATLABwith functions capable of handling the

multivariable input-output scheme. All the results and examples except for example1.1.2.1 were obtained with the MIMO Toolbox and were corroborated with the

bibliography.

April, 2006

Installation

The installation is straightforward just copy the directory Mimotools and add the path

to the MATLABsearch path.

See path, in the MATLABdocumentation for more information.

Requirements

The MIMO Toolbox was created in Matlab 7.1 (R14) SP3, and requires the Symbolic

and Control Toolboxes.

Contact

Oskar Vivero

[email protected]


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Contents

1. Theory behind the functions

1.1.SISO Systems

1.1.1. Feedback Basic Concepts

1.1.2. Nyquists Stability Criterion

1.2.MIMO Systems

1.2.1. Poles and Zeros of a MIMO System

1.2.1.1. Smith-McMillan Transformation1.2.2. Stability of MIMO Systems

1.2.2.1. Generalized Nyquists Stability Criterion

1.2.3. Treating a MIMO System with SISO techniques

1.2.3.1. Coupling degree and pairings of inputs and outputs

1.2.3.2. Nyquists Arrays and Gershgorin Bands

1.2.3.3. Relative Gain Array (RGA)

1.2.3.4. Individual Channel Design (ICD)

2.

Function Reference

2.1.The Symbolic Transfer Function

2.2.

Function Description

2.2.1.

tf2sym2.2.2. sym2tf

2.2.3. ss2sym

2.2.4. smform

2.2.5. rga

2.2.6. nyqmimo

2.2.7. m_circles

2.2.8. icdtool

2.2.9. gershband

2.2.10.arrowh

3. References

1

1

1

2

4

4

47

7

10

10

10

11

12

14

14

14

1415

16

16

17

17

19

19

20

21

22


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1. Theory behind the functions

The aim of this chapter is to introduce the MIMO control theories so that one can

understand both, the algorithm behind each function and its proper use. This chapter is

only intended to provide a brief description of such theories and its recommended that

the user refers to the bibliography listed at the end of this document.

1.1 SISOSystems

1.1.1 Feedback Basic Concepts

Assuming a lineal process that is time-invariant whose behavior is defined by lineal

differential equations with constant coefficients:

1 2 1 2y a y a y b u b u

+ + = +

where ( )y t is the output signal and ( )u t is the input signal, its possible to obtain a

transfer function by applying the Laplaces Transform

( )

( ) ( )

( )

( )1 2

2

1 2

Y s N s b s bG s

U s D s s a s a

+= = =

+ +

( )

( )

If 0 then is defined as a zero.

If then is defined as a pole.

Z Z Z

P P P

s s G s s

s s G s s

= =

= =

If ( )G s is rational, usually ( )D s determines the dynamic characteristics of the system,unless there exist cancellations between ( )N s and ( )D s .

Let ( )H s and ( )G s be two transfer functions

The stability of the system in a closed-loop configuration is given by ( ) ( )1 G s H s+ ifand only if there is no cancellation of instabilities. For any design, its possible to

verify its stability by finding the singularities of ( )CLG s . If any of the singularities is

located in2

+ or near the imaginary axis, its almost impossible to determine the

modifications needed on either ( )G s or ( )H s to avoid the location of the singularities.

The Nyquists stability criterion provides a tool for solving the problem

( ) ( )

( ) ( )1CL

G sG s

G s H s=

+


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1.1.2 Nyquists Stability Criterion

The system in a closed-loop configuration is stable if and only if the trajectory of the

Nyquist diagram of ( ) ( )G j H j from < < surrounds the point ( )1,0 in a

counter-clockwise direction as much times as ( ) ( )G s H s has unstable poles.

Theorem suppose that a function ( )f z is meromorphic in a simply connected domain

D, and that C is a simple closed positively oriented contour in D such that ( )f z does

not contain any singularities. Then

( )

( )

'1

2f f

f zN dz Z P

i f z= =

where Nis the winding number,f

Z is the number of zeros inside the contour andf

P

is the number of poles inside the contour.

Example 1.1.2.1

The image of the circle of radius 2 centered at the origin under ( ) 2f z z z= + is the

curve ( ) ( ) ( ) ( ) ( )( ), 4cos 2 2cos , 4sin 2 2sing x y t t t t = + + . Note that the curve ( ),g x y winds up twice around the origin. We check this by computing

( )

( )

0 1

0 1

2 2

'1 ; Singulatiries at 0 and 1

2

2 1 2 1

Res Res 2

C

z z

f zN z z

i f z

z z

N z z z z

= = =

+ + = + = + +

Having defined N, its important to define a useful contour for the stability analysis.

Its possible to know from the root locus analysis and the time response that the

unstable poles are at the right side of the S-plane. Since the zeros of the open loop

system are the poles of the closed loop system, well focus on finding the unstable zeros

through Nyquist frequency analysis. The contours that well consider are:


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Example 1.1.2.2 [4] pp. 620

Draw the Nyquist contour and diagram of ( )( )( ) ( )

500

1 3 10G s

s s s=

+ + +

Once the stability of a system has been defined through the Nyquists diagram, therobustness of the system can be defined by two quantities, the gain and phase margins.

Gain margin ( )MG - the change of gain in open loop needed to obtain a phase shift at

180 that turns the system unstable.

Phase margin ( )M - the phase shift in open loop needed to turn the system unstable

with a unit gain.


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1.2 MIMOSystems

The basic description of a multivariable system is through a transfer function matrix

(TFM), whose elements ,i jg represent the i-est output with the j-est input. The

elements,i j

g are individual transfer functions.

1.2.1 Poles and zeros of a MIMO system

The poles and zeros of multivariable systems can be defined in several (not all

equivalent) ways, but the definitions that yield the most significant consequences are

given by [1]:

The zeros of a transfer function matrix ( )H s , are the roots of the (nonzero) numerator

polynomials ( ){ }i s in the Smith-McMillan form of ( )H s . The Smith-McMillan form

allows us to give a physical interpretation of the zeros. If Zs is a zero, then the Smith-McMillan form of ( )H s will lose rank at Zs s= .

The poles of a transfer function matrix ( )H s , are the roots of the denominator

polynomials in the Smith-McMillan form of ( )H s .

1.2.1.1 Smith McMillan Transformation

Given a rational matrix ( )H s

( ) ( )

( )

N sH s

d s

=

where ( )d s is the monic least common multiple of the denominators of ( )H s

Then ( ) ( ) ( )d s H s N s= is a polynomial matrix, so that we can write,

( ) ( ) ( ) ( ) ( ) ( )1 2d s H s N s U s s U s= =


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where the ( ){ }iU s are unimodular matrices and ( )s is in Smith form

( ) ( ) ( ) ( )

( )

( )

( )

( )

( )

( )

( )

1 1

1 2

i

i i

i

s sU s H s U s diag

d s d s

s s

d s s

= =

=

where ( ) ( ){ },i is s are coprime, 1, ,i r=

and r is the (normal) rank of ( )H s

Then we can write

( ) ( ) ( ) ( )1 2H s U s M s U s=

where ( )M s is the Smith-McMillan transformation of ( )H s given by

( )

( )

( )0

0 0

i

i

sdiag

M s s

=

Smith Form

For any p m polynomial matrix ( )P s we can find elementary row and column

operations, or corresponding unimodular matrices ( ) ( ){ },U s V s such that

( ) ( ) ( ) ( )U s P s V s s=

where

( )

( )

( )

( )

0

0

0 0 0

the (normal) rank of

i

r

s

ss

r P s

=

=

and the ( ){ }i s are unique monic polynomials obeying a division property

( ) ( )1 , 1, , 1i is s i r + =

Moreover, if( ) ( )the gcd of all minors ofi s i i P s =

then

( ) ( )

( ) ( )0

1

, 1i

i

i

ss s

s

= =

The matrix ( )s is called the Smith form of ( )P s


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Example.1.2.1.1 [1] pp. 444-446

Find the poles and zeros of

( )( ) ( )

( )

( ) ( )

2

2 2 2 2

11

1 2 1 1

s s sG s

s s s s s s

+ = + + + +

SolutionGiven

( )( )

( )1

G s N sd s

=

where

( ) ( ) ( ){ } ( ) ( )

( )

( )

( ) ( )

2 2 2 2

2

2 2

1 2 1 2

1

1 1

d s lcm s s s s

s s s

N s s s s s

= + + = + +

+

= + +

Finding ( )s in Smith form:

( )

( ) ( ) ( ) ( )( )

( ) ( ) ( )( )( ) ( )

( ) ( ) ( ) ( )

0

2 2 2

1

23

2

1 1

22

2 2

1

gcd , 1 , 1 , 1

gcd 1 2

1 2

s

s s s s s s s s s

s s s s

s s s

s s s s s

=

= + + + =

= + +

= =

= = + +

Therefore

( )

( )

( ) ( ) ( )

2 2

2

00 1 2

00 02

i

i

ss

diag s sM s s

s

s

+ + = =

+

The poles are defined as

( ) ( ) ( ) ( ) ( ) ( )[ ]

2 2

1 2 1 2 2 0

1 1 2 2 2

p s s s s s s

p

= = + + + ==

The zeros are defined as

( ) ( ) ( )

[ ]

3

1 2 0

0 0 0

z s s s s

z

= = =

=


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1.2.2 Stability of MIMO Systems

Having defined a tool in order to obtain the poles and zeros of a MIMO system, its

necessary to define if the system is stable. This can be achieved by the generalized

Nyquists stability criterion, which is an adaptation from the Nyquists stability criterion

for SISO systems.

1.2.2.1 Generalized Nyquists Stability Criterion

Let ( )G s be a rational TFM, assuming that it has no cancellations between poles and

zeros. Let Kbe a compensator with a negative feedback loop, that is

I ;K k k=

Let the ( )det I KG s+ have P poles and Zzeros in the right hand plane (RHP), then

like in SISO systems

( ) ( )arg det I 2KG s Z P + =

where arg is the change of phase given by salong the Nyquists contour. In order to

obtain stability in a closed loop, it is required that the diagram mapped by

( )det I KG s+

surrounds the origin Ptimes.

As a difference from the SISO case, the Nyquist diagram of each kof interest must be

obtained.

If ( )i s is an eigenvalue of ( )G s , then ( )ik s is an eigenvalue of ( )KG s , and

therefore, ( )1 ik s+ is an eigenvalue of ( )I KG s+ . Then

( ) ( )det I 1 ii

KG s k s+ = +

and

( ) ( )arg det I arg 1 ii

KG s k s + = +

Therefore, the stability in a closed loop system can be inferred by the number of wind

ups to the point ( )1,0 given by the Nyquist diagram of ( )ik s . The Nyquist diagram

of ( )i s are known as the characteristic graphs of ( )G s .


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Theorem If ( )G s has right hand plane poles (RHPP), P, given by the Smith-McMillan

transformation, then the closed loop with negative feedback is stable if and only if the

characteristic graphs of ( )KG s surround the point ( )1,0 P times in a counter-

clockwise direction, assuming that there was no cancellations of instabilities.

Example 1.2.2.1 [2] pp. 61Find the values of 0k> in order to keep ( )G s from becoming unstable

( )( )( )

11

6 21.25 1 2

s sG s

ss s

=

+ +

Solution

Its easy to verify that the open loop poles of ( )G s are [ ]1 2P = , therefore, in

order to maintain the closed loop system stable, we must ensure that the number of

surroundings of the characteristic graphs of ( )G s is equal to zero.Assuming 1k= , then

( )

( )( )

( )( )

2

1

2

2

det I 0

2 2 3 24 1

5 3 2

2 2 3 24 1

5 3 2

KG s

s s

s s

s s

s s

= =

+ + + + = = + + +

From the eigenvalues of ( )G s , its possible to obtain the Generalized Nyquist Diagramwhen 1k =


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From the diagram its possible to obtain the critical points in which we can calculate the

values of k, in order to keep the system stable. Knowing that the critical points are in

0.8 and 0.4 , therefore

1 11.25 , 2.5

0.8 0.4k k

< = > =

This can be proven by setting the values of kequal to 1.25 and 2.5 and obtaining the

Nyquist Diagram.


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1.2.3 Treating a MIMO system with SISO techniques

1.2.3.1 Coupling degree and pairing of inputs and outputs

Given the coupling degree of a MIMO system, its possible to apply certain SISO

techniques for designing controllers. The degree of coupling can be found by several

techniques, such as the Nyquists arrays and the Gershgorin bands, the relative gain

array (RGA), and the individual channel design (ICD). In some cases, it is possible tocross couple inputs and outputs in order to obtain a less coupled system.

1.2.3.2 Nyquists Arrays and Gershgorin bands

The Nyquists array of a TFM ( )G s , is an array of graphs, where the i, j-est graph is the

Nyquist Diagram of ,i jg .

Gershgorin Theorem let Z a n n complex matrix. Then, the eigenvalues of Z fall

on the union of circles with center at,i i

z with radius

,1

m

i jj

j i

Z=

and on the circles with center at,j j

z with radius

,1

m

i ji

i j

Z=

Gershgorin Bands

Over the Nyquists diagram of ( ),i ig s , on each point, we super impose a circle of

radius

( ) ( ), ,1 1

orm m

i j j ii i

i j i j

g j g j = =

The bands obtained in this way are known as the Gershgorin Bands.

By the Gershgorin theorem its possible to know that the bands trap the unions of theNyquist diagram. More over, its possible to demonstrate that the bands occupy

different regions, therefore there will be as many Nyquists diagrams trapped in a region

as many Gershgorin bands are there.

Then, by counting the number of wind ups that the Gershgorin bands do around the

point ( )1,0 , its possible to determine the stability of the MIMO system.


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If the Gershgorin bands are thin and exclude the origin, it is said that ( )G s is

diagonally dominant which can be interpreted as a decoupled system.

Example 1.2.3.1 [3] pp. 657

Find the Gershgorin bands of

( )2

2 2

2 0.1

13 2

0.1 6

2 1 5 6

ss sG s

s s s s

++ +

= + + + +

From the graphs above, its possible to observe that the system is stable and highly

decoupled. Therefore, it can be managed as two independent SISO systems with small

disturbances due to the small coupling degree.

1.2.3.3 Relative Gain Array (RGA)

To measure the degree of coupling or interaction in a system, the concept of relative

gain array can be used. For an arbitrary n n matrix A, it is defined as

( ) ( )1RGA .*T

A A A=

The RGA matrix has a number of interesting properties

The sum of the elements of any row or column is always 1 RGA is independent of any scaling

The sum of the absolute values of all elements in ( )RGA A is a good measure of

As true condition number, i.e., the best condition number that can be achieved

in the family 1 2D AD , where iD are diagonal matrices.

Permutation of the rows or columns of A leads to permutations of the

corresponding rows or columns of ( )RGA A .


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( )

( )

12 21

11 22

1

i ii

i

i ii

g gs

g g

k gh s

k g

=

=+

The MSF ( )s is of great importance inside the ICD analysis framework, since it is

capable of [6]

Determining the dynamical characteristics between each input and each output

It has an interpretation in the frequency domain

Its magnitude quantifies the amount of coupling between the channels

It can determine the transmittance zeros of the system from ( )1 s

( ) 1s = determines the non-minimum phase conditions

Its closeness to the point ( )1,0 its a key point in determining the robustness ofthe system

Robustness Conditions

In order to obtain a design that provides a channel that is robust and stable, the

following conditions should be satisfied

1. ( )s should not be close to the point ( )1,0 for all

2. ( ) ( )is h s shall be robust

3. ( ) ( )i iik s g s shall be robust

The interaction between the discarded inputs and outputs can be observed from

( ) ( )

( ) ( )

( ) ( )

1

1

ij

i j j

jj i

g sY s h s R s

g s C s=

+


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2. Function Reference

The aim of this chapter is to give a brief description of the functions used in the MIMO

toolbox. For the users that are new to the Matlab environment it is recommended to

review the getting started documentation.

2.1 The Symbolic Transfer Function

The Control Toolbox in Matlab posses an object class that describes a transfer function.

This model representation is numerical and its sensitive to floating point errors due to

arithmetic operations such as inversion. Also, from the LTI definition, the transfer

function class cannot handle nonlinearities such as square roots, trigonometric

functions, etc, since its only a rate in polynomial representation. Given such problems,

a symbolic conversion for transfer function has been developed, and its key to some of

the functions inside the MIMO toolbox. This conversion enables the user to handle the

transfer function and its operations as an algebraic problem, simplifying the

computational difficulties that may arise for the users without a computer sciencebackground.

2.2 Function Description

2.2.1 tf2sym

SyntaxG_sym = tf2sym(G)

Description

tf2sym performs the conversion from a numeric to a symbolic representation of atransfer function. The function is capable to deal with both SISO and MIMO models.

To be able to differentiate the type of transfer function, the symbolic transfer function is

represented with the Laplace operator p instead of s.

Example 1>> G=tf([1 2 3],[1 5 6 7])

>> G_sym=tf2sym(G);

>> pretty(G_sym)

2

3 + p + 2 p

-------------------

3 2p + 5 p + 6 p + 7

Example 2>> g11=tf([1 2],[1 2 1]);

g12=tf([1 -1],[1 5 6]);

g21=tf([1 -1],[1 3 2]);

g22=tf([1 2],[1 1]);

G=[g11 g12; g21 g22];

g=tf2sym(G);

pretty(g)


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[ 2 + p -1 + p ]

[ -------- ---------------]

[ 2 (p + 3) (2 + p)]

[ (p + 1) ]

[ ]

[ -1 + p 2 + p ]

[--------------- ----- ]

[(2 + p) (p + 1) p + 1 ]

2.2.2 sym2tf

SyntaxG = sym2tf(G_sym)

Description

sym2tf performs the conversion from a symbolic to a numeric representation of a

transfer function. As tf2sym, this function is capable of dealing with both SISO and

MIMO models. This conversion cannot handle nonlinearities since the numeric transfer

function class tf is not capable of dealing with the nonlinearities.

Example 1>> syms p

>> G_sym = (p^2 + 2*p + 3)/(p^3 + 5*p^2 + 6*p + 7);

>> G=sym2tf(G_sym)

Transfer function:

s^2 + 2 s + 3

---------------------

s^3 + 5 s^2 + 6 s + 7

Example 2>> g11=(p + 2)/(p^2 + 2*p + 1);

g12=(p - 1)/(p^2 + 5*p + 6);

g21=(p - 1)/(p^2 + 3*p + 2);

g22=(p + 2)/(p + 1);

g=[g11 g12; g21 g22];

G=sym2tf(g)

Transfer function from input 1 to output...

s + 2

#1: -------------

s^2 + 2 s + 1

s - 1

#2: -------------

s^2 + 3 s + 2

Transfer function from input 2 to output...s - 1

#1: -------------

s^2 + 5 s + 6

s + 2

#2: -----

s + 1


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2.2.3 ss2sym

SyntaxG = ss2sym(A,B,C,D)

Description

ss2symperforms the conversion from a state space representation to a symbolic transfer

function.

Example>> A = [0 1 0 0 0; 0 0 1 0 0; -2 -5 -4 0 0; 0 0 0 0 1; 0 0 0 -3 -4];

B = [0 0; 0 0; 1 0; 0 0; 0 1];

C = [1 2 1 9 3; 14 9 1 1 1];

D = [0 1; 0 0];

g=ss2sym(A,B,C,D);

pretty(g)

[ 1 p + 4]

[ ----- -----]

[ 2 + p p + 1]

[ ][ p + 7 1 ]

[-------- -----]

[ 2 p + 3]

[(p + 1) ]

2.2.4 smform

Syntax[M,poles,zeros] = smform(G)

Description

[M,poles,zeros] = smformcomputes the Smith-McMillan transformation, the poles

and zeros of a MIMO TFM. The algorithm obeys directly the theory specified on thesection 1.2.2.1 of this document by establishing an active link with the Maple kernel.

Example>> g11=tf([1 1],[1 3 2]);

g12=tf([1 4],[1 1]);

g21=tf([1 7],[1 2 1]);

g22=tf([1 2],[1 5 6]);

G=[g11 g12 ; g21 g22];

[M,poles,zeros]=smform(G);

>> pretty(M)

[ 1 ]

[------------------------ 0 ]

[ 2 ]

[(p + 3) (p + 1) (2 + p) ]

[ ]

[ 4 3 2 ]

[ p + 15 p + 86 p + 203 p + 167]

[ 0 --------------------------------]

[ p + 1 ]


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>> poles

poles =

-3.0000

-2.0000

-1.0000

-1.0000 + 0.0000i

-1.0000 - 0.0000i

>> zeros

zeros =

-5.3101 + 2.5439i

-5.3101 - 2.5439i

-2.1899 + 0.1460i

-2.1899 - 0.1460i

2.2.5 rga

SyntaxA=rga(G)

Description

rgacomputes the Relative Gain Array of a MIMO TFM. The algorithm obeys directly

the theory specified on the section 1.2.3.3

Example>> g11=tf([1 2],[1 2 1]);

g12=tf([1 -1],[1 5 6]);

g21=tf([1 -1],[1 3 2]);

g22=tf([1 2],[1 1]);

G=[g11 g12; g21 g22];

A=rga(G)

A =

1.0213 -0.0213

-0.0213 1.0213

2.2.6 nyqmimo

Syntaxnyqmimo(G)

Description

nyqmimocomputes the Nyquists Diagram based on the type of transfer function at theinput. nyqmimodistinguishes between transfer functions with singularities at the origin,

proper, strictly proper and strictly improper transfer functions, based on that structure, it

maps the Nyquists Diagram according to the Nyquists Contour. nyqmimo is capable

of computing the Nyquists Diagram for SISO and the Generalized Nyquists Diagram

for MIMO systems.


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Example 1>> G=tf([1 25],[1 5 3 -9 0]);

nyqmimo(G)

Example 2

>> den=1.25*conv([1 1],[1 2]);g11=tf([1 -1],den);

g12=tf([1 0],den);

g21=tf([-6],den);

g22=tf([1 -2],den);

G=[g11 g12;g21 g22];

nyqmimo(G)

2.2.7 m_circles

Syntaxm_circles

Description

m_circlessuperimposes the M circles of magnitude -2dB to 20dB. This function is

useful to measure the gain margin on a Nyquists Diagram.


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Example>> G=tf(1,[1 2 1]);

>> nyqmimo(G)

Verifying for sigularities on the origin...

Setting frequency range...

Mapping...

Plotting...

>> m_circles

2.2.8 icdtool

Syntaxicdtool(G)

Descriptionicdtool is a GUI designed to help the user in the task of designing controllers for a

2 2 MIMO system with under the Individual Channel

Design Scheme.

By default, icdtool calculates ( )s , when just loaded, icdtool will not load the

default controller. The controller will only be loaded and all the proper calculations

performed after the user has clicked on the UPDATE CONTROLLERS button. If the

user doesnt know entirely the behavior of the system, it is recommended that they load

a unitary gain proportional controller, in that way, the user will be able to analyze both,

( )s and the diagonal elements of the system, ( ) ( )1,1 2,2,g s g s . It is possible to load a

proportional controller by clearing the poles and zeros entries.


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2.2.9 gershband

Syntaxgershband(G) Gershgorin bands of G

gershband(G,v) Gershgorin bands and Nyquist Array of G

Description

gershbandcomputes the Gershgorin Bands of a n n MIMO system along with the

Nyquists Array.

Example>> g11=tf([1 2],[1 2 1]); g12=tf([1 -1],[1 5 6]);

g21=tf([1 -1],[1 3 2]); g22=tf([1 2],[1 1]);

G=[g11 g12; g21 g22];

gershband(G,v);


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2.2.10 arrowh

SyntaxArrowh(x,y,color,size,location)

Description

arrowhdraws a solid arrowhead in the current plot

Author: Florian Knorn

Email: [email protected], [email protected]

Homepage: http://www.florian-knorn.com/


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MIMO Toolbox

3. References

[1] Kailath, Thomas, Linear Systems, Prentice Hall, 1980, pp. 390-392, 443-450[2] Maciejowski, J.M., Multivariable Feedback Design, Addison-Wesley, 1989,

pp. 37-71

[3] Goodwin, Graham C., Graebe, Stefan F., Salgado, Mario E., Control System

Design, Prentice Hall, 2001, pp. 653-671

[4] Nise, Norman S., Sistemas de Control para Ingeniera, CECSA, 2002, pp. 614-

635

[5] Ogata, Katsuhiko, Ingeniera de Control Moderna, Prentice-Hall, 2003, 523-

566

[6] Carlos E. Ugalde-Loo, Eduardo Licaga-Castro, Jess Licaga-Castro, 2x2

Individual Channel Design MATLAB Toolbox Proceedings of the 44th IEEE

CDC, pp. 7603-7608

[7] J. Liceaga, E. Liceaga and L. Amzquita, Multivariable Gyroscope Control byIndividual Channel Design, Control Systems Society CCA, pp. 110-115, 2005

[8] E, Liceaga-Castro and J. Liceaga-Castro, Submarine depth control by

individual channel design, Proceedings of the 37th

IEEE CDC, e, pp. 3183-

3188, 1998

[9] J. Licaga-Castro, C. Ramrez-Espaa and E. Licaga-Castro, GPC control

design for a Temperature and Humidity Prototype using ICD anlisis,

Submitted for publication.

MIMO Toolbox

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